UC-NRLF 


LECTURE  NOTES 


ON 


CRYSTALLOGRAPHY 


BY 


HORACE  BUSHNELL  PATTON,  Ph.  D. 

$ 

Professor  of  Geology  and   Mineralogy  at  the  State   School 

of  Mines,  Golden,  Colorado. 


7EB1 

v- 


PUBLISHED  BY  THE  AUTHOR, 
GOLDEN,  COLORADO. 


COPYRIGHT,     1896, 

BY 
HORACE   B.    PATTON. 


NEWS   PRINTING  COMPANY, 
1896. 


PREFACE. 

The  difficulty  of  presenting  the  subject  of  crystallogra- 
phy introductory  to  a  course  in  mineralogy,  with  only  a 
limited  time  at  one's  disposal,  has  been  realized  by  all 
instructors  in  this  branch.  Even  the  best  text  books  fail  to 
present  the  subject  in  so  clear  a  light  as  to  do  away  with 
the  necessity  of  lecturing.  The  main  difficulty  with  the 
best  treatises  on  this  subject,  when  put  in  the  hands  of  the 
student,  is  their  length  .and  multiplicity  of  details. 

These  lecture  notes  are  not  intended  to  form  a  treatise 
on  crystallography.  They  were  originally  prepared,  not  for 
publication,  but  for  use  in  the  class  room,  supplemented  by 
lectures,  crystal  models  and  natural  crystals.  In  the  very 
limited  time  allotted  to  the  study  of  mineralogy  it  was  found 


ERRATA. 

PAGE  8.     Reverse  the  two  signs    >.    <  .   in  the  table  of  forms. 

PAGE  11.     Between  the  second  and  third  lines  of  italics  at  the  top  insert — 
all  the  faces  that  lie  wholly  within. 

PAGE  11.     Center  of  page.  Should  read— Trisoctahedron gives  Tetra- 
gonal tristetrahedron. 


alogy. 

On  account  of  the  brevity  of  these  notes  the  interleaved 
blank  pages  have  been  found  very  useful,  and  they  will 
undoubtedly  prove  serviceable  in  case  any  one  else  should 
wish  to  use  these  notes  as  a  basis  for  instruction. 

STATE  SCHOOL  OF  MINES,  GOLDEN,  COLO.,  April,  1896. 


NEWS  PRINTING  COMPANY, 


PREFACE. 

The  difficulty  of  presenting  the  subject  of  crystallogra- 
phy introductory  to  a  course  in  mineralogy,  with  only  a 
limited  time  at  one's  disposal,  has  been  realized  by  all 
instructors  in  this  branch.  Even  the  best  text  books  fail  to 
present  the  subject  in  so  clear  a  light  as  to  do  away  with 
the  necessity  of  lecturing.  The  main  difficulty  with  the 
best  treatises  on  this  subject,  when  put  in  the  hands  of  the 
student,  is  their  length  .and  multiplicity  of  details. 

These  lecture  notes  are  not  intended  to  form  a  treatise 
on  crystallography.  They  were  originally  prepared,  not  for 
publication,  but  for  use  in  the  class  room,  supplemented  by 
lectures,  crystal  models  and  natural  crystals.  In  the  very 
limited  time  allotted  to  the  study  of  mineralogy  it  was  found 
necessary  to  omit  everything  not  absolutely  essential  to  a 
clear  and  logical  presentation  of  the  subject  of  crystallogra- 
phy. 

Requests  for  copies  of  these  lecture  notes  by  other 
parties  has  led  the  author  to  hope  that  they  may  prove  of 
service  to  some  outside  his  own  class  room.  No  attempt  is 
here  made  to  enter  upon  many  features  of  crystals,  such  as 
physical  and  optical  properties,  crystal  aggregates,  twinning, 
etc.  These  omissions  are  made  for  the  reason  that  such 
features  are  usually  clearly  presented  in  almost  any  miner- 
alogy. 

On  account  of  the  brevity  of  these  notes  the  interleaved 
blank  pages  have  been  found  very  useful,  and  they  will 
undoubtedly  prove  serviceable  in  case  any  one  else  should 
wish  to  use  these  notes  as  a  basis  for  instruction. 

STATE  SCHOOL  OF  MINES,  GOLDEN,  COLO.,  April,  1896. 


CRYSTALLOGRAPHY. 

In  the  inorganic  world  bodies  are  met  with  that  are 
homogeneous  throughout;  others,  again,  that  consist  of  two 
or  more  kinds  of  homogeneous  substances  ;  and  still  others 
that  are  composed  of  many  distinct  grains  or  particles  of 
one  kind  of  substance.  The  first  are  termed  Minerals,  the 
last  two  Rocks. 

A  Mineral,  therefore,  may  be  defined  as  an  inorganic, 
natural,  homogeneous  body. 

A  mineral  represents  a  definite  chemical  compound, 
and  its  formation  is  controlled  by  the  same  laws  that  con- 
trol the  formation  of  all  chemical  compounds. 

These  natural  laws  of  growth  express  themselves  in 
many  ways,  e.  g.,  through  outward  form  and  through  vari- 
ous physical  and  optical  properties  of  the  mineral.  These 
different  manifestations,  however,  vary  with  and  thus  char- 
acterize the  chemical  combination. 

As  an  illustration  of  these  properties  we  may  take  what 
is  called  cleavage.  Just  as  a  block  of  wood  cleaves  in  cer- 
tain directions,  dependent  on  the  grain  of  the  wood,  i.  e.,  on 
the  arrangement  of  the  wood  cells,  so,  many  minerals  are 
said  to  cleave  in  certain  definite  directions.  And  this  min- 
eral cleavage  is  likewise  due  to  a  sort  of  grain  in  the  min- 
eral produced  by  the  arrangement  of  the  molecules.  To 
illustrate,  rock-salt  cleaves  always  in  three  directions  at 
right  angles  to  each  other,  while  crystallized  carbonate  of 
lime  cleaves  in  three  directions  that  make  an  angle  of  about 
105°  to  each  other. 

Another,  and  more  important,  outward  manifestation  of 
the  laws  of  mineral  growth  is  seen  in  the  natural  external  form. 
There  are  three  ways  in  which  minerals  are  usually  formed, 
namely,  through  solution,  through  fusion  and  through  sub- 
limation. But  in  any  case,  unless  interfered  with  by  some 
external  agency,  the  mineral  particles  in  forming  are  usually 
found  to  be  bounded  on  all  sides  by  plane  surfaces  and  to 


—  4  — 

assume  definite  polygonal  forms,  which  forms  vary  with  the 
chemical  composition  of -the  mineral,  but  are  constant  for 
any  one  definite  mineral  species. 

A  mineral  thus  bounded  entirely  or  partly  by  natural 
plane  surfaces  is  termed  a  CRYSTAL. 

Crystallography,  as  far  as  these  lecture  notes  are  con- 
cerned, may  be  defined  as  a  study  of  the  geometrical  rela- 
tionships of  these  natural  crystal  faces. 

Before  taking  up  the  study  of  crystal  forms  it  is  neces- 
sary to  have  clearly  in  mind  a  few  fundamental 


DEFINITIONS. 

A  COMMON  SYMMETRY  PLANE  is  any  plane  that  divides 
a  crystal  body  into  two  symmetrical  halves;  and  a  body  may 
be  said  to  be  symmetrically  divided  when  the  following  test 
holds  true.  Every  perpendicular  to  the  dividing  plane,  if 
extended  in  both  directions  to  the  surface  of  the  crystal,  must 
come  out  in  points  equally  distant  from  the  plane  and  simi- 
larly located  on  the  crystal.  If  the  symmetry  plane  were 
conceived  to  be  replaced  by  a  mirror,  the  image  reflected 
by  the  half-crystal  in  front  of  the  mirror  would  appear  to 
coincide  in  position  with  the  half-crystal  behind  the  mirror. 

A  SYMMETRY  Axis  is.  any  line  (or  the  direction}  perpen- 
dicular to  a  symmetry  plane. 

A  PRINCIPAL  SYMMETRY  PLANE  is  a  plane  in  whicJi  lie 
two,  sometimes  three,  interchangeable  symmetry  axes.  (These 
interchangeable  symmetry  axes  are  not  the  axes  of  this 
plane,  but  of  other  symmetry  planes  at  right  angles  to  this 
plane). 

Two  axes  are  said  to  be  interchangeable  when,  by  turn- 
ing the  crystal,  you  can  bring  one  axis  into  the  position  oc- 
cupied by  the  other  without  thereby  producing  any  appar- 
ent change  in  the  position,  form  or  direction  of  the  crystal. 

A  PRINCIPAL  SYMMETRY  Axis  is  any  line  (or  the  direc- 
tion) perpendicular  to  a  principal  symmetry  plane. 

FACES.  A  crystal  is  bounded  by  plane  surfaces  called 
faces. 


EDGES.  The  line  of  intersection  between  two  faces  is 
called  an  edge. 

ANGLES.  These  are  of  two  sorts.  An  inter  facial  angle  is 
the  plane  angle  formed  by  the  intersection  of  two  faces.  A 
crystal  angle  is  the  solid  angle  formed  by  the  intersection  of 
three  or  more  faces. 

SIMILAR  EDGES  AND  ANGLES  are  those  formed  by  the 
intersection  of  the  same  number  of  planes,  similarly  placed. 

Upon  bringing  together  all  known  crystal  forms  it  is 
seen  that  they  may  all  be  grouped  into  six  crystal  systems 
that  differ  from  each  other  by  the  number  and  kind  of  the 
symmetry  planes  present.  Below  will  be  found  grouped  in 
accordance  with  the  presence  of  principal  and  common 
symmetry  planes  the 


SIX    CRYSTAL    SYSTEMS. 


I.  WITH  3  PRINCIPAL  SYMMETRY  PLANES. 

(1)  with  six  common  symmetry  planes. 

II.  WITH  1  PRINCIPAL  SYMMETRY  PLANE. 

(2)  a,  with    four    common     symmetry 

planes  arranged  in  two  pairs  of 
perpendicular  planes. 

(3)  b,  with      six     common     symmetry 

planes  arranged  in  two  groups  of 
three  each,  making  angles  of  60° 
to  each  other. 

III.  WITH  NO  PRINCIPAL  SYMMETRY  PLANE. 

(4)  a,  with     three   common    symmetry 

planes  making  right  angles  with 
each  other. 

(5)  b,  with  one  common  symmetry  plane. 

(6)  c,  with  no  symmetry  plane   what- 

ever. 


Isometric  System. 


Tetragonal  System. 


Hexagonal  System. 


Orthorhombic  System. 


Monoclinic  System. 
Triclinic  System. 


USE  OF  CRYSTAL  AXES. 

In  addition  to  the  symmetrical  arrangement  of  the 
faces  it  has  been  noticed  that,  however  numerous  may  be 
the  faces  on  a  crystal,  they  bear  further  definite  relationships 
to  each  other  in  that,  within  certain  limits,  the  facial  angles 
appear  to  be  fixed.  The  determination  of  these  facial  angles 


—  6  — 

is  an  important  part  of  the  study  of  crystallography,  and  for 
this  purpose  it  has  been  found  desirable  to  refer  all  the 
faces  of  a  crystal  to  three  (sometimes  four)  arbitrarily 
chosen  fixed  lines.  If  we  can  determine  the  inclinations 
which  the  different  faces  make  to  these  fixed  lines  we  can 
thereby  determine  their  mutual  inclinations  to  each  other. 
These  fixed  lines  are  called  Crystal  Axes. 

Theoretically  these  crystal  axes  might  be  chosen  mak- 
ing any  desired  angles  to  each  other,  provided  the  angles 
be  known,  but  for  the  sake  of  simplicity  it  is  found  desirable 
to  choose  them  so  that  they  may  bear  some  definite  relation- 
ship to  the  symmetry  of  the  crystal. 

Therefore,  we  have  the  following 

GENERAL  RULE  FOR  CHOOSING  CRYSTAL  AXES. 

In  all  the  six  systems  the  crystal  axes  are  chosen  to 
coincide  with  symmetry  axes,  as  far  as  this  is  possible ;  a 
principal  symmetry  axis  always  being  given  the  preference. 

PARAMETERS. 

All  the  crystal  axes  are  supposed  to  meet  and  cross  each 
other  at  the  geometric  centre  of  the  crystal.  In  ascertain- 
ing the  inclinations  of  the  faces  to  each  other  by  means  of 
the  crystal  axes  we  conceive  both  faces  and  axes  to  be  ex- 
tended indefinitely,  or  until  the  plane  of  any  face  under  con- 
sideration cuts  all  three  axes,  or  until  it  is  evident  that  the 
plane  never  will  cut  the  axis.  The  relative  distances  at 
which  the  plane  of  a  face  cuts  the  three  axes,  measured 
from  the  centre  of  the  axial  cross,  are  called  the  parameters 
of  the  face.  Experience  has  shown  that  the  value  of  a 
parameter  is  either  infinity  or  some  small  rational  quantity. 
The  parameter  value  infinity  is  expressed  thus,  oo,  and  in- 
dicates that  the  face  is  parallel  to  the  axis. 

CRYSTAL  FORM. 

In  studying  crystals  in  any  one  system  it  will  be 
noticed  that,  if  we  start  with  any  face  with  known  parameter 
values,  in  order  that  the  law  of  symmetry  of  the  system  may 


7 

be  fulfilled  there  must  also  occur  a  definite  number  of  other 
faces  with  exactly  the  same  parameter  values  as  has  the 
face  we  start  with.  In  speaking  of  crystal  form,  then,  we 
use  the  term  in  a  technical  sense  and  mean  all  the  faces 
taken  together  that  are  required  to  complete  the  symmetry 
of  the  system.  Crystal  form,  therefore,  may  be  defined  as 
the  sum  of  all  those  crystal  faces,  which  the  symmetry  of  a 
crystal  demands,  when  one  of  them  is  present. 

LAW  OF  SYMMETRY. 

Both  ends  of  either  a  symmetry  or  a  crystal  axis,  and  the 
ends  of  all  interchangeable  axes  must  be  cut  by  the  same  num- 
ber of  crystal  faces,  similarly  placed. 

This  is  a  fundamentally  important  law,  and  holds  good 
with  certain  modifications  to  be  noted  beyond. 


ISOMETRIC     SYSTEM. 

This  system  has  three  principal  symmetry  planes  at 
right  angles  to  each  other ;  also  six  common  symmetry 
planes  lying  intermediate  between  and  diagonal  to  the  prin- 
cipal symmetry  planes,  and  making  with  each  other  angles 
of  60°,  90°  and  120°. 

The  three  principal  symmetry  planes  divide  the  crystal 
body  into  eight  solid  parts  called  octants. 

The  three  principal  symmetry  axes  are  interchange- 
able and,  in  accordance  with  the  rule,  are  chosen  as  the 
three  crystal  axes.  The  law  of  symmetry,  therefore,  in  this 
system  demands  that  the  six  ends  of  the  three  interchange- 
able axes  should  be  cut  by  the  same  number  of  planes, 
similarly  placed,  i.  e.,  the  six  ends  of  the  crystal  axes  must 
come  out  in  similarly  located  points.  A  little  thought  will 
also  show  that  all  eight  octants  must  contain  the  same 
number  of  crystal  faces. 

Crystals  in  this  system  show  an  equal  development 
in  three  directions. 


—  8  — 


CRYSTAL  FORMS  IN  THE  ISOMETRIC  SYSTEM. 

In  determining  crystal  forms  it  will  suffice  if  we  can 
determine  for  a  form  how  any  one  of  its  faces,  if  extended, 
would  cut  the  three  crystal  axes,  inasmuch  as  all  the  other 
faces  of  the  same  form  must  cut  the  axes  at  the  same  dis- 
tances. We  can  indicate  how  the  plane  of  a  face  cuts  the 
three  axes  by  means  of  the  three  parameter  values  expressed 
in  the  form  of  a  ratio.  Thus,  a  :  nb  :  me,  where  the  letters 
a,  b  and  c  represent  the  three  crystal  axes,  and  the  coeffi- 
cients m  and  n  give  the  parameter  values.  In  the  isometric 
system,  as  all  three  axes  are  interchangeable,  they  must  all 
be  treated  alike,  and  it  is  not  necessary  to  distinguish  be- 
tween a,  b  and  c.  We  will  designate  all  three,  therefore,  by 
the  same  letter,  a.  The  parameter  values  expressed  in  the 
form  of  a  ratio  may  be  called  the  SYMBOL  of  a  crystal 
face,  and  also  of  the  form  to  which  the  face  belongs. 

In  the  table  given  below  will  be  found  all  the  possible 
variations  of  crystal  symbols  in  the  isometric  system,  so  ar- 
ranged as  to  make  it  quite  evident  that  no  variations  are 
left  out. 

ISOMETRIC    HOLOHEDRAL    FORMS. 


Three  axes  cut  alike.  __ 

a 

:  a  : 

a 

Octahedron 

8  Faces    ///       ^ 

Two 
axes 
cut 
alike 

Two  axes  cut  at 
distance  j^the 
other  __  __ 

a 
a 

:  a  : 
:  a  : 

ma 
oca 

Trisoctahedron 
Dodecahedron 

24  Faces   *  h  '      '' 
12  Faces    /  o  i 

Two  axes  cut  at 
distance*^  the 
other  ^ 

a 
a 

:  ma  : 
:  oca  : 

ma 
oca 

Trapezohedron 
Hexahedron 

24  Faces     /?// 
6  Faces     /  00 

Three  axes  cut  unlike  _ 

a. 
a 

:  ma  : 

:  ma  : 

na 
cca 

Hexoctahedron 
Tetrahexahedron 

48  Faces     '7  /<c  / 
24  Faces     h  a  / 

Of  these  seven  possible  isometric  forms,  one  has  six 
faces,  one  eight,  one  twelve,  three  twenty- four,  and  one 
forty- eight.  The  hexahedron  with  six  faces  has  its  faces 
parallel  to  the  principal  symmetry  planes.  The  dodecahe- 
dron with  twelve  faces  has  its  faces  parallel  to  the  six  com- 
mon symmetry  planes. 


—  9  — 

The  twenty-four  sided  forms  are  the  most  confusing. 
One  of  them  may  be  readily  determined  by  noting  that  its 
faces  are  each  parallel  to  a  crystal  axis.  In  studying  these,  i 
as  in  all  other  forms,  first  find  the  three  principal  symmetry 
axes,  which  are  chosen  as  crystal  axes.  Place  the  crystal 
so  that  one  axis  is  vertical  and  two  horizontal,  one  of  which 
runs  from  front  to  back  and  the  other,  necessarily,  from 
right  to  left.  For  the  two  other  twenty- four-sided  forms, 
namely,  the  trapezohedron  and  the  trisoctahedron,  the  fol- 
lowing rule  is  very  useful,  provided  that  it  is  a  question  of 
deciding  between  these  two  forms  only  :  The  trisoctahe- 
dron has  its  faces  so  arranged  in  each  octant  that  an  edge 
runs  from  the  centre  of  the  octant  out  to  each  axis,  while 
the  trapezohedron  has  a  face  running  from  the  centre  of  the 
octant  out  to  each  axis.  (This  rule  will  also  apply  in  case 
of  combined  forms  mentioned  below.) 

COMBINATION  OF  FORMS. 

A  crystal  in  this  system  may  have  any  one  of  these 
seven  forms,  or  it  may  have  two  or  more,  or  even  all  seven 
developed  at  the  same  time.  These  forms,  of  course,  must 
then  mutually  modify  each  other,  and  when  so  occuring 
cannot  be  expected  to  have  the  same  shape  as  in  the  un- 
modified forms.  On  the  other  hand,  the  inclination  of  the 
faces  to  the  axes  cannot  be  in  the  least  changed  by  such 
modification,  so  that  each  form  may  be  determined  by  its 
symbol  as  readily  as  in  the  uncombined  forms. 

RULE  FOR  COMBINATION  OF  FORMS. 

First.  All  the  faces  of  the  same  crystal  form  on  the 
same  crystal  have  the  same  shape  and  size. 

Second.  Thei  e  are  as  many  different  crystal  forms  on  a 
crystal  as  there  are  different  kinds  of  faces. 

(The  above  rules  hold  only  for  crystal  models  or  for 
perfectly  developed  natural  crystals). 

The  following  terms  are  used  in  describing  combined 
forms  : 

A  face  is  said  to  replace  an  edge  when  it  cuts  the  crystal 
parallel  to  the  replaced  edge. 


—  10  — 


An  edge  or  angle  is  truncated  when  it  is  replaced  by  a 
face  making  equal  angles  with  the  two  adjacent  faces. 

An  edge  is  bevelled  ivhen  it  is  replaced  by  two  faces  that 
are  equally  inclined  to  the  faces  making  the  replaced  edge. 


HOLOHEDRAL,  HEMIHEDRAL  AND  TETARTOHEDRAL  FORMS. 

Each  of  the  above  described  seven  forms  have  all 
faces  that  the  fully  developed  symmetry  of  the  system 
demands,  hence  they  are  called  Holohedral  Forms.  This 
term  is  used  to  distinguish  them  from  other  forms  that 
have  only  half  of  the  faces  developed  that  the  symmetry  of 
the  system  demands.  Such  forms  are  designated  Hemi- 
hedral  Forms.  Still,  others  having  but  one-fourth  of  the 
faces  developed  are  called  Tetartohedral  Forms. 

There  are  many  ways  in  which  half  the  faces  of  a  crys- 
tal form  might  be  selected,  and  nature  actually  does  choose 
more  than  one  way,  but  the  faces  are  not  selected  without 
regard  to  rule,  for  in  all  cases  there  must  hold  true  the  fol- 
lowing 

LAW    OF    SYMMETRY    OF    HEMIHEDRAL    AND    TETARTOHEDRAL 
FORMS. 

The  ends  of  all  similar  (holohedral}  symmetry  axes  must 
be  cut  by  the  same  number  of  crystal  faces,  similarly  placed. 

The  following  rule  is  important  as  to 

NUMBER  OF  HEMIHEDRAL  AND  TETARTOHEDRAL 
FORMS.  Every  holohedral  form  has  a  corresponding  hemiJie- 
dral  or  tetartoJiedral  form. 

The  above  law  and  rule  can  only  hold  true  when  the 
parts  to  be  developed  are  obtained  by  cutting  the  holohedral 
forms  into  sections  by  means  of  symmetry  planes.  The 
kind  of  hemihedrism  will  Depend  upon  the  choice  of  these 
symmetry  planes.  There  are  thus  three  possible  hemi- 
hedrisms  in  the  isometric  system.  The  first  obtained  by 
cutting  the  holohedral  forms  by  means  of  the  principal 
symmetry  planes;  the  second,  by  using  for  this  purpose 
the  common  symmetry  planes ;  the  third,  by  using  both 
sets  of  symmetry  planes. 


—  11  — 

I.  INCLINED  HEMIHEDRAL  FORMS  IN  THE  ISOMETRIC  SYSTEM. 

Inclined  hemihedral  forms  may  be  conceived  to  be  de- 
veloped by  suppressing  on.  each  of  the  seven  holohedral  forms 

OLlt   lhfif«*.**.*k£t  i-C*  "Wly   f^Ai*       •>•      '  j>  JT          7      I    1      J        7 

four  alternating  octants  \obiained  by  dividing  the  holohedral 
form  by  the  three  principal  symmetry  planes],  while  the  remain- 
ing faces  are  developed. 

By  this  process  we  suppress  faces  on  one  side  of  a 
principal  symmetry  plane  and  develop  on  the  other  side. 
This  necessarily  destroys  the  principal  symmetry  plane. 

Inclined  hemihedral  forms,  therefore,  may  be  distin- 
guished by  the  fact  that  they  have  the  six  common  symmetry 
planes,  but  not  the  principal  symmetry  planes  of  the  isometric 
system. 

The  following  forms  result  from  the  application  of  the 
above  law. 

Octahedron gives  Tetrahedron with    4  faces 

Trapezohedron gives  Trigonal  tf igtetrahed^on with  12  faces 

Trisoctahedron gives  Tetragonal  S8B68(18HlSSL  _  _  with  12  faces 

Hexoctahedron gives  Hextetrahedron with  24  faces 

Hexahedron gives  Hexahedron with    6  faces 

Dodecahedron gives  Dodecahedron with  12  faces 

Tetrahexahedron... gives  Tetrahexahedron with  24  faces 

The  first  four  of  the  above  forms  occur  with  just  half 
as  many  faces  as  have  the  corresponding  holohedral  forms, 
but  the  last  three  occur  with  all  the  faces  present  in  the 
holohedral  forms.  They  may  appear  to  be  holohedral,  but 
are  really  hemihedral  forms.  Hemihedral  forms,  there- 
fore, cannot  always  be  distinguished  from  holohedral  by  the 
number  of  their  faces.  The  reason  why  these  three  forms 
occur  with  all  the  faces  of  the  corresponding  holohedral 
forms  is  evident  when  we  consider  that  they  have  infinity 
in  their  symbol,  i.  e.,  each  face,  being  parallel  to  an  axis, 
must  lie  in  two  adjacent  octants  and  cannot,  therefore,  be 
suppressed  according  to  the  rule  for  inclined  hemihedral 
forms. 

II.  PARALLEL  HEMIHEDRAL  FORMS  IN  THE  ISOMETRIC  SYSTEM. 

Parallel  hemihedral  forms  may  be  conceived  to  be  devel- 
oped by  suppressing  on  each  of  the  seven  holohedral  forms  all 


-12- 


the  faces  that  lie  wholly  within  twelve  alternating  parts  ob- 
tained by  dividing  the  holohedral  form  by  means  of  the  com- 
mon symmetry  planes,  while  the  remaining  faces  are  developed. 

By  this  process  we  destroy  planes  on  one  side  of  a 
common  symmetry  plane  and  develop  those  on  the  other 
side.  This  necessarily  destroys  the  common  symmetry 
planes. 

Parallel  hemihedral  forms,  therefore,  may  be  distin- 
guised  by  the  fact  that  they  have  the  three  principal  symmetry 
planes  but  not  the  six  common  symmetry  planes  of  the  isomet- 
ric system. 

The  following  forms  result  from  the  application  of  the 
above  law  : 

Tetrahexahedron gives  Pentagonal  dodecahedron __ with  12  faces 

Hexoctahedron gives  Didodecahedron with  24  faces 

Octahedron .gives  Octahedron with    8  faces 

Hexahedron gives  Hexahedron with    6  faces 

Dodecahedron gives  Dodecahedron with  12  faces 

Trapezohedron gives  Trapezohedron with  24  faces 

Trisoctahedron gives  Trisoctahedron with  24  faces 

In  this  case  there  are  only  two  forms  that  occur  with 
half  as  many  faces  as  do  the  corresponding  holohedral 
forms.  The  last  five  forms,  though  apparently  holohedral, 
are  really  hemihedral. 

.  The  reason  why  five  forms  occur  with  as  many  faces  in 
the  parallel  hemihedral  division  as  in  holohedral  is  similar 
to  that  given  for  the  inclined  hemihedral  forms.  Each  face 
is  so  placed  that  it  lies  in  at  least  two  adjacent  parts,  and 
therefore  cannot  be  suppressed. 

III.    6YROIDAL  HEMIHEDRAL  FORMS  IN  THE  ISOMETRIC  SYSTEM 

Gyroidal  hemihedral  forms  may  be  conceived  to  be  devel- 
oped by  suppressing  on  each  of  the  seven  hoi oliedral  forms  the 
faces  lying  wholly  within  twenty-four  alternating  parts 
obtained  by  cutting  the  Jwlohedral  forms  into  forty-eight  sections 
by  means  of  botli  sets  of  symmetry  planes,  while  the  remain- 
ing faces  are  developed, 

By  this  process  all  the  symmetry  planes  are  destroyed, 
and  gyroidal  hemihedral  forms  may  be  distinguished  by  this 
tact. 


—  13  — 

As  there  is  but  one  holohedral  form  with  forty-eight 
faces,  this  can  evidently  be  the  only  form  whose  faces  lie 
wholly  within  the  forty-eight  parts  into  which  the  nine  sym- 
metry planes  cut  a  crystal.  Therefore,  the  hexoctahedron 
can  be  the  only  form  giving  a  new  hemihedral  form.  This 
is  called  the  pentagonal  icositetrahedron.  There  are  two  of 
these,  distinguished  as  right  and  left-handed,  differing  from 
each  other  only  as  a  right-handed  glove  differs  from  a  left- 
handed. 

Gyroidal  hemihedral  forms  are  very  rare  and  are  of  no 
importance  from  a  practical  point  of  view. 

TETARTOHEDRAL  FORMS  IN  THE  ISOMETRIC  SYSTEM. 

Tetartohedral  forms  are  also  very  rare  in  the  isometric 
system.  As  a  description  of  these  forms  is  not  thought  to 
be  in  accord  with  the  object  of  these  notes,  the  reader  is 
referred  to  larger  treatises  on  crystallography  and  mineral- 
ogy, The  principles  upon  which  tetartohedral  forms  are 
conceived  to  be  developed  will  be  found  set  forth  in  these 
Lecture  Notes  under  the  hexagonal  system. 

GENERAL    REMARKS    ON    HEMIHEDRAL    AND    TETARTOHEDRAL 
FORMS. 

No  substance  crystallizes  both  holohedral  and  hemi- 
hedral or  tetartohedral,  and  two  kinds  of  hemihedral  forms, 
or  hemihedral  and  tetartohedral  forms  are  never  found  on 
the  same  crystal.  We  do  find,  however,  apparantly  holohe- 
dral, (but  really  hemihedral  forms)  occurring  with  other  hem- 
ihedral forms. 

The  symbols  of  hemihedral  forms  are  the  same  as  those 
of  holohedral  forms,  but  they  are  written  in  the  form  of  a 
fraction  with  2  for  a  denominator.  Similarly,  tetartohedral 
forms  have  the  denominator  4. 

Holohedral  forms  really  give  two  hemihedral  forms 
which  are  usually  exactly  alike,  except  in  position,  and  are 
designated  as  positive  and  negative.  Thus,  the  octahedron 
gives  a  +  tetrahedron  and  a  —  tetrahedron.  The  same 
holds  true  for  all  the  systems. 


—  14  — 


TABLE  GIVING  THE  SYMBOLS  OF  THE  FORMS  IN  THE  ISO- 
METRIC SYSTEM  AS  USED  BY  WEISS,  NAUMANN,  DANA 
AND  MILLER. 


WEISS 

NAUMANN. 

DANA. 

MILLER. 

Octahedron            

B 

a 

a, 

o 

I 

(Ill) 

Hexahedron       

a 

ooa 

on  a 

CO  O  CO 

H 

(100) 

Dodecahedron  _  __ 

a 

a 

on  a 

ooO 

i 

(110) 

Trisoctahedron  .  .  

a 

a 

ma 

mO 

m 

(hhl) 

Trapezohedron 

a 

ma 

ma 

m  O  m 

m-m 

Ml) 

Tetrahexahedron 

a 

ma 

on  a 

co  O  n 

i-n 

(hkO) 

Hexoctahedron 

a 

ma 

na 

m  O  n 

m-n 

(hkl) 

TETRAGONAL  SYSTEM. 

In  this  system  there  is  but  one  principal  symmetry 
plane  and  four  common  symmetry  planes  all  at  right  angles 
to  the  principal  symmetry  plane.  These  four  common 
symmetry  planes  occur  as  two  pairs  of  right  angled  planes, 
each  pair  standing  intermediate  between  or  45°  inclined  to 
the  other  pair. 

There  is,  therefore,  one  direction,  namely,  that  of  the 
principal  symmetry  axis  independent  of  and  distinguished 
from  the  others.  The  two  common  symmetry  axes  of  each 
pair  are  interchangeable  with  each  other,  but  not  with  the 
principal  symmetry  axis,  nor  with  the  common  symmetry 
axes  of  the  other  pair. 

In  choosing  the  three  crystal  axes  we  will,  in  accordance 
with  the  general  ride,  select  the.  principal  symmetry  axis  for 
one  crystal  axis,  and  for  the  two  other  crystal  axes  we  will 
select  either  one  of  the  two  pairs  of  common  symmetry  axes. 

In  studying  the  crystal  forms  it  is  customary  to  place 
the  principal  symmetry  plane  horizontal.  The  principal 
symmetry  axis,  therefore,  becomes  the  vertical  crystal  axis. 
The  two  other  selected  crystal  axes  are  horizontal,  one  of 
them  being  placed  from  front  to  back.  The  two  horizontal 
axes,  being  interchangeable,  are  both  designated  by  the  let- 
ter a,  and  the  vertical  axis  by  the  letter  c. 

The  vertical  axis  c  can  never  be  equal  to  the  horizontal 
axis  a,  and  the  ratio  between  c  and  a  can  never  be  a  rational 


-15- 


+ 


V*v     jJ»  4 

quantity.     In  the  symbols  given  below,  however,  the  c"6^: 


efficients  before  c  and  a  are  always  rational  quantities  and 

may  also  be  equal. 

As  ic  can  never  be  equal  to  la,  m  may  become  equal 
without  changing  the  character  of  the  form. 


to 


HOLOHEDRAL   TETRAGONAL    FORMS. 


a 

a 

me 

Direct  pyramid 

8  Faces 

First  Order  

a 

a 

OOC 

Direct  prism 

4  Faces 

a 
a 

na 

na 

me 

ccc 

Ditetragonal  pyramid 
Ditetragonal  prism___ 

16  Faces 
8  Faces 

a 

ooa 

me 

Indirect  pyramid. 

8  Faces 

Second  Order.  _ 

a 

ooa 

occ 

Indirect  prism    __ 

4  Faces 

oca 

ooa 

c 

Basal  pinacoid 

2  Faces 

COMBINATION  OF  FORMS.  Only  in  case  of  one 
of  the  three  pyramids  can  a  single  crystal  form  entirely 
bound  a  crystal.  The  other  forms  can  occur  only  in  com- 
bination. 

The  basal  pinacoid  and  the  prisms  of  the  first  and 
second  orders  have  no  variable  in  their  symbol.  They  can 
occur  therefore  but  once  on  a  crystal.  All  the  other  forms 
are  variable  and  can  occur  an  indefinite  number  of  times  on 
the  same  crystal. 

The  forms  in  this  system  can  be  readily  determined 
without  the  aid  of  the  symbols  by  means  of  the  following 


RULES  FOR  DETERMINING  TETRAGONAL  FORMS. 

First.  A  face  which  is  parallel  to  both  horizontal  axes 
is  the  basal  pinacoid. 

Second.  A  face  whose  plane  cuts  all  three  axes  be- 
longs to  a  pyramid.  If  it  cuts  the  two  horizontal  axes  alike 
it  belongs  to  a  direct  pyramid;  if  unlike,  to  a  ditetragonal 
pyramid. 

Third.  A  face  whose  plane  cuts  the  vertical  axis  and 
is  parallel  to  one  of  the  horizontal  axes  belongs  to  an  in- 
direct pyramid. 


ff  «*  <M  r  rr 


—  16  — 

Fourth.  A  face  parallel  to  the  vertical  axis  belongs  to 
a  prism.  If  the  plane  of  the  face  cuts  both  horizontal  axes 
alike  it  belongs  to  a  direct  prism;  if  unlike,  to  a  ditetragonal 
prism.  If  the  plane  cuts  one  horizontal  axis  and  is  parallel 
to  the  other,  it  belongs  to  an  indirect  prism. 

Fifth.  If  two  or  more  pyramids,  or  a  prism  and  pyra- 
mid, make  with  each  other  horizontal  edges^  they  are  of  the 
same  order. 


HEMIHEDRAL  FORMS  IN  THE  TETRAGONAL  SYSTEM. 

There  are  three  possible  kinds  of  hemihedrism  in  this 
system,  and  these  are  closely  analogous  with  those  in  the 
isometric  system  in  their  method  of  development. 

I.  SPHENOIDAL  HEMIHEDRISM.  As  is  the 
case  with  inclined  hemihedral  forms,  these  are  devel- 
oped by  dividing  the  holohedral  forms  by  means  of  the 
principal  symmetry  plane,  and  by  the  two  symmetry  planes 
containing  the  horizontal  axes.  These  give  us  octants, 
which  are  alternately  suppressed  and  developed. 

From  the  pyramid  of  the  first  order  there  is  obtained  the 
sphenoid,  This  is  equivalent  to  the  tetrahedron,  from 
which  it  differs  in  that  the  two  horizontal  edges  are  differ- 
ent from  the  four  other  edges. 

From  the  ditetragonal  pyramid  there  is  developed  the 
tetragonal  scalenohedron,  which  does  not  closely  resemble 
any  isometric  form. 

These  two  are  the  only  new  forms,  all  the  others  being 
externally  identical  with  the  corresponding  holohedral 
forms. 

II.    PYRAMIDAL  HEMIHEDRISM. 

To  develop  these  forms  we  conceive  each  of  the  holo- 
hedral forms  to  be  cut  into  eight  parts  by  means  of  both 
sets  of  common  symmetry  axes. 

The  ditetragonal  pyramid  gives  us  a  new  form  called 
the  pyramid  of  the  third  order,  which  differs  in  no  respect 


—  17  — 

from  the  pyramids  of  the  first  and  second  orders,  except  that 
it  occupies  a  position  unsymmetrical  with  reference  to  the 
common  symmetry  planes  and  to  the  other  forms. 

Similarly  the  ditetragonal  prism  gives  the  prism  of  the 
third  order  with  similar  relationships  to  the  symmetry  planes 
and  to  the  other  forms. 

All  the  other  forms  occur  as  though  they  were  holo- 
hedral. 

III.    TRAPEZOHEDRAL  HEMIHEDRISM. 

To  develop  these  forms  we  conceive  each  of  the  holohe- 
dral  forms  to  be  cut  by  means  of  all  five  symmetry  planes 
into  sixteen  parts,  each  corresponding  in  position  to  that  of 
the  ditetragonal  pyramid. 

The  ditetragonal  pryamid,  obviously,  is  the  only  form 
that  can  give  a  new  form,  It  is  called  the  tetragonal  trap- 
ezohedron. 

There  are  no  known  natural  minerals  crystallizing  in 
this  form,  although  certain  organic  salts  are  known  to 
belong  to  this  division. 


TETARTOHEDRAL  FORMS  IN  THE  TETRAGONAL  SYSTEM. 

As  no  minerals  are  absolutely  known  to  crystallize 
tetartohedral  in  this  system,  the  possible  tetartohedral 
divisions  are  omitted.  They  are  exactly  analogous  to  those 
that  will  be  found  described  under  the  hexagonal  system. 

TABLE   OF    SYMBOLS    IN    THE   TETRAGONAL    SYSTEM. 


WEISS 

NAUMANN 

DANA 

MILLER 

Direct  Dyramid 

a  • 

a  :  me 

mP 

m 

(hhl) 

Indirect  pyramid 

8    ' 

QO  a  :  me 

mPoo 

m-i 

(hOl) 

Ditetragonal  pyramid  _  _  _ 

a  : 

na  :  me 

mPn 

m-n 

(hkl) 

Direct  prism  

a  : 

a  :  ooc 

OOP 

I 

(110) 

Indirect  prism     _ 

a,  : 

QO  a  :  QOC 

oo  Poo 

i-i 

(100) 

Ditetragonal  prism  .  _ 

a,  : 

na  :  ooc 

oo  Pn 

i-n 

(hkO) 

Basal  pinacoid 

on  a 

:  GO  a  :  me 

OP 

o 

(001) 

—  18  — 

HEXAGONAL   SYSTEM. 

This  system  is  very  closely  analogous  to  the  tetragonal 
system.  All  the  forms  exactly  correspond  with  those  in 
that  system,  except  that  the  faces  here  occur  in  multiples  of 
three  instead  of  in  multiples  of  two. 

In  the  hexagonal  system  there  is  one  principal 
symmetry  plane  at  right  angles  to  six  common  symmetry 
planes.  The  latter  are  grouped  in  two  triplets.  The  faces 
in  each  triplet  stand  at  an  angle  of  60°  to  each  other,  but 
the  two  triplets  alternate  with  each  other,  making  angles 
of  30°. 

The  direction  of  the  principal  symmetry  axis  is  evi- 
dently independent  of  and  readily  distinguished  from  all 
others.  This  is  chosen  as  the  vertical  crystal  axis.  It 
would  be  possible  to  select  from  among  the  common 
symmetry  axes  two  others  that  are  at  right  angles  to  each 
other,  but  as  these  would  not  be  equivalent  or  interchange- 
able they  would  not  at  all  represent  the  symmetry  of  the 
system.  It  has  been  found,  therefore,  more  desirable  to 
select  in  this  case  three  horizontal  axes  coinciding  with 
either  one  of  the  two  triplets  of  common  symmetry  axes. 

The  three  horizontal  crystal  axes  are  thus  interchange- 
able and  make  with  each  other  angles  of  60°.  The  crystal 
is  so  placed  that  one  of  these  axes  runs  from  right  to  left, 

The  vertical  axis  is  designated  the  c  axis,  and  the  in- 
terchangeable horizontal  axes  the  a  axes. 


HOLOHEDRAL  HEXAGONAL  FORMS. 


First  ) 
Order_  \ 

a 
a 

a 
a 

co  a 
co  a 

me 

COC 

Pyramid  of  the  1st  order. 
Prism  of  the  1st  order  

12  faces 
6  faces 

pa 
Da 

a 
a 

na 
na 

me 

CO  C 

Dihexaszonal  pyramid  
Dihexagonal  prism 

24  faces 
12  faces 

Second  ) 
Order  .  \ 

2a 
2a 

a 
a 

2a 

2a 

me 

CO  C 

Pyramid  of  the  2d  order.  _ 
Prism  of  the  2d  order  

12  faces 
6  faces 

Goa  : 

coa 

oca 

c 

Basal  pinacoid  

2  faces 

—  19  — 

COMBINATION  OF  FORMS. 

As  is  the  case  in  the  tetragonal  system  the  pyramids 
are  the  only  forms  that  can  wholly  bound  a  crystal.  The 
other  forms  can  occur  only  in  combination.  The  basal 
pinacoid  and  the  prisms  of  the  first  and  second  order, 
having  no  variable  in  their  symbol,  can  occur  but  once, 
while  the  other  forms  with  variable  symbols  can  occur  an 
indefinite  number  of  times  on  the  same  crystal. 

The  forms  in  this  system  may  be  readily  determined 
without  the  aid  of  symbols  by  means  of  the  following 

RULES  FOR  DETERMINING  HEXAGONAL  FORMS. 

First.  A  face  which  is  parallel  to  the  horizontal  axes  is 
the  basal  pinacoid. 

Second.  A  face  whose  plane  cuts  the  vertical  axis 
obliquely  belongs  to  a  pyramid.  If  the  plane  cuts  two  of 
the  lateral  axes  equally  and  is  parallel  to  the  third  the 
pyramid  is  of  \htfirst  order.  If  one  horizontal  axis  is  cut 
at  unity  and  the  two  others  at  twice  the  distance  the  pyra- 
mid is  of  the  second  order.  If  the  three  horizontal  axes  are 
all  cut  unequally  the  pyramid  is  dihexagonal. 

Third.  A  face  parallel  to  the  vertical  axis  belongs  to  a 
prism.  If  the  plane  cuts  two  horizontal  axes  equally  and 
is  parallel  to  the  third  the  prism  is  of  the  first  order.  If  one 
horizontal  axis  is  cut  at  unity  (perpendicularly)  and  the  two 
others  at  twice  the  distance  the  prism  is  of  the  second  order. 
If  the  three  horizontal  axes  are  all  cut  unequally  the  prism 
is  dihexagonal. 

Fourth.  If  two  or  more  pyramids  or  a  prism  and  a 
pyramid  make  with  each  other  horizontal  edges  they  are  of 
the  same  order. 

HEMIHEDRAL  FORMS  IN  THE  HEXAGONAL  SYSTEM. 

The  hemihedral  forms  in  the  hexagonal  system  are  of 
greater  importance  than  those  of  any  other  system,  with  the 
possible  exception  of  the  isometric  system,  and  are,  there- 
fore, treated  at  greater  length. 

There  are  three  possible  kinds  of  hemihedrism  in  this 
system,  corresponding  to  those  given  for  the  tetragonal 
system. 


—  20  — 

I.     RHOMBOHEDRAL  HEMIHEDRISM. 

This  kind  of  hemihedrism  may  be  conceived  to  be 
developed  by  cutting  each  of  the  seven  holohedral  forms 
into  twelve  parts  (called  dodecants)  by  means  of  the  princi- 
pal symmetry  plane  and  by  one  set  of  common  symmetry 
planes,  namely,  by  the  three  planes  in  which  the  horizontal 
axes  have  been  chosen ;  and  by  suppressing  all  the  faces 
that  lie  wholly  within  six  alternating  dodecants,  while  all 
the  remaining  faces  are  developed. 

In  case  of  five  of  the  holohedral  forms,  namely,  basal 
pinacoid,  first  and  second  order  prisms,  second  order  pyra- 
mid and  dihexagonal  prism,  no  one  face  lies  wholly  within 
one  dodecant,  therefore,  no  face  can  be  suppressed  in  accord- 
an  ce  with  the  above  stated  law.  These  five  forms,  there- 
fore, have  hemihedral  forms  exactly  like  the  corresponding 
holohedral  forms,  and  as  such  they  may  occur  in  combina- 
tion with  the  apparently  as  well  as  really  hemihedral  forms 
given  below. 

Two  of  the  holohedral  forms  give  new  half  forms. 
The  dihexagonal  pyramid  gives  the  scalenohedron.  The 
pyramid  of  the  first  order  gives  the  rliombohedron. 

The  scalenohedron  is  a  twelve  sided  form  that  may  be 
distinguished  from  the  hexagonal  pyramid  by  the  following 
facts  :  First,  the  lateral  edges  are  not  horizontal,  but  zig- 
zagged. Second,  the  edges  running  to  the  vertex  are  alter- 
nately sharper  and  blunter.  The  sharper  edge  above  lies 
over  the  blunter  edge  below  and  vice  versa. 

The  rhombohedron  is  a  six  sided  form,  with  three  faces 
above  and  three  below.  The  following  properties  may  be 
noted:  First,  the  three  edges  running  to  the  vertex  are 
equal,  but  are  different  from  the  zig  zag  lateral  edges. 
Second,  the  upper  faces  do  not  lie  over  the  under  faces,  but 
alternate  with  them. 

In  general,  rhombohedral  hemihedral  forms  may  be 
recognized  by  the  fact  that  they  have  only  three  symmetry 
planes,  namely,  the  three  planes  lying  intermediate  between 
the  crystal  axes. 


—  21  — 

II.  PYRAMIDAL  HEMIHEDRISM. 

This  kind  of  hemihedrism  may  be  conceived  to  be 
developed  by  cutting  each  of  the  seven  holohedral  forms 
into  twelve  parts,  or  dodecants,  by  means  of  the  two  sets  of 
common  symmetry  planes ;  and  by  suppressing  all  the 
faces  that  lie  wholly  within  six  alternating  dodecants,  while 
the  remaining  faces  are  developed. 

Here,  too,  we  find  that  five  of  the  holohedral  forms 
cannot  give  new  half  forms  inasmuch  as  they  do  not  have 
their  faces  lying  wholly  within  the  dodecants.  These  forms 
are  basal  pinacoid,  pyramid  and  prism  of  the  first  order  and 
pyramid  and  prism  of  the  second  order.  These  five  forms, 
therefore,  if  they  occur  at  all,  must  occur  with  all  their 
faces. 

The  two  other  forms  are : 

Dihexagonal  pyramid,  gives  pyramid  of  the  third  order, 
with  twelve  faces. 

Dihexagonal  prism,  gives  prism  of  the  third  order,  with 
six  faces. 

The  third  order  pyramid  and  prism,  differ  in  no  respect 
from  the  first  or  second  order  pyramid  and  prism  except 
in  position.  They  lie  unsymmetrical  with  reference  to  the 
horizontal  axes,  i.  e.,  they  cut  the  horizontal  axes  unequally. 
When  they  occur  in  combination  with  other  prisms  and 
pyramids,  their  unsymmetrical  position  is  easily  noted. 

In  general,  the  pyrimidal  hemihedral  forms  may  be 
recognized  by  the  fact  that  they  have  only  the  principal  sym- 
metry plane  present. 

III.  TRAPEZOHEDRAL  HEMIHEDRISM. 

These  forms  may  be  conceived  to  be  developed  by  cut- 
ting each  of  the  seven  holohedral  forms  into  twenty-four 
parts  by  means  of  all  seven  symmetry  planes;  and  by  sup- 
pressing all  the  faces  that  lie  wholly  within  twelve  alternat- 
ing parts,  while  all  the  remaining  faces  are  developed. 

In  this  case  there  is  only  one  form  whose  faces  lie 
wholly  within  these  twenty-four  parts,  namely,  the  dihexago- 
nal  pyramid.  Therefore,  this  is  the  only  form  that  can  give  a 


—  22  — 

new  hemihedral  form.     This  form  is  called  the  hexagoi 
trapezohedron.     It   consists    of  six    faces    above,   that 
neither  exactly  above  nor  exactly  alternating  with  the  six 
below. 

In  general,  this  form  may  be  recognized  by  the  absence 
of  all  symmetry  planes,  combined  with  an  hexagonal 
arrangement  of  the  faces. 

TETARTOHEDRAL  FORMS  IN  THE  HEXAGONAL  SYSTEM. 

Tetartohedral  or  quarter  forms  may  be  conceived  to  be 
developed  from  the  holohedral  forms  by  the  simultaneous  or 
successive  application  to  each  of  these  forms  of  two  kinds  of 
hemihedrism.  As  there  are  three  kinds  of  hemihedrism,  there 
are  three  possible  combinations  to  be  considered.  In  every 
tetartoliedral  form,  however t  the  following  must  hold  true  : 
The  opposite  ends  of  a  (holohedral)  symmetry  axis  and  ends 
of  all  interchangeable  (holohedral)  symmetry  axes  must  be  cut 
by  the  same  number  of  planes  similarly  placed. 

We  will  now  see  whether  the  above  condition  can  be 
fulfilled  by  combining,  first,  the  rhombohedral  and  trapezo- 
hedral  hemihedrisms;  second,  the  rhombohedral  and  pyra- 
midal hemihedrisms  ;  third,  the  pyramidal  and  trapezohe- 
dral  hemihedrisms. 

If  a  tetartohedral  form  is  possible  it  can  certainly  be 
developed  on  the  most  general  form,  the  dihexagonal  pyra- 
mid. We  will  apply  the  test  to  this  form  for  the  three  cases. 
The  figures  below  represent  the  twenty-four  faces  of  the 
dihexagonal  pyramid  arranged  in  an  upper  and  a  lower  bank 
to  correspond  with  the  position  of  the  faces  on  the  holohe- 
dral form.  (This  method  is  taken  from  Prof.  Groth's  well 
known  work  on  Physical  Crystallography.)* 

The  faces  that  would  be  suppressed  by  the  application 
of  the  rhombohedral  hemihedrism  alone  are  cancelled  by 
diagonal  lines,  thus,  /;  those  that  would  be  suppres- 
sed by  the  application  of  the  trapezohedral  hemihedrism  are 
cancelled  by  diagonal  lines,  thus,  \ ;  and  those  that  would 


*  P.  Groth.      Physikalische  Krystallographie.      Leipzig.      Wilhelm 
Engelmann. 


—  28  — 

be  suppressed  through  the  pyramidal  hemihedrism  are  can- 
celled by  horizontal  lines. 

First.  Suppression  through  rhomb ohedral  and  trapezo- 
hedral  hemihedrisms. 

Upper  faces  \    2    X   XX    6    X  X  X    10    X     J/ 
Lower  faces  f  X    %   \  XX    7  XX   X  _  U  .  K, 

Leaving  faces  2,  6,  10  above,  and  3,  7,  11  below   to   be 
developed. 

Second.  Suppression  through  rhombohedral  and  pyra- 
midal hemihedrisms. 

Upper  faces   4-2^>T-^6^^-^10     &    yt 
Lower  faces  if  #  -8-    4    7^  X  ~*     S^Uftt     12 

Leaving  faces  2,  6,   10  above,  and  4,  8,  12   below  to   be 

developed. 

Third.  Suppression  through  pyramidal  and  trapezohe- 
dral  hemihedrisms. 

Upper  faces  ^    f\    4   ^     6   ^'8    \    10     *t     12 
Lower  faces  -i-  _\-3-\-6-X-^X-e-^tt     H, 

Leaving  faces  2,  4,  6,  8,  10,  12  above,  and  none  below. 

It  is  evident  that  the  last  case  does  not  give  a  form 
that  fulfills  the  above  conditions,  but  these  conditions  are 
found  to  be  fulfilled  in  the  first  two  cases.  We  have,  there- 
fore, two  possible  kinds  of  tetartohedrism.  First.  The  tra- 
pezohedral  tetartohedrism,  developed  by  the  simultaneous 
application  of  the  rhombohedral  and  the  trapezohedral 
hemihedrisms,  and  second,  the  rhombohedral  tetartohedrism, 
developed  by  the  simultaneous  application  of  the  rhombo- 
hedral and  the  pyramidal  hemihedrisms. 

I.    TRAPEZOHEDRAL  TETARTOHEDRISM. 

If  this  method  be  applied  to  the  seven  holohedral  forms 
it  is  found  that  two  of  them  can  give  no  new  form,  as  none 
of  their  faces  can  be  suppressed.  They  are  the  basal  pina- 
coid  and  the  prism  of  the  first  order. 


24:  

Basal  pinacoid  gives Basal  pinacoid. 

Pyramid  of  the  first  order  gives Rhombohedron. 

Prism  of  the  first  order  gives Prism  of  the  first  order. 

Pyramid  of  the  second  order  gives Trigonal  pyramid. 

Prism  of  the  second  order  gives Trigonal  Prism. 

Dihexagonal  pyramid  gives Trigonal  trapezohedron. 

Dihexagonal  prism  gives Ditrigonal  prism. 

The  tetartohedral  rhombohedron  cannot  be  distinguished 
from  the  hemihedral  rhombohedron.  It  occurs  both  posi- 
tive and  negative. 

The  trigonal  prism  has  three  faces  that  would  give  in 
cross-section  an  equilateral  triangle. 

The  trigonal  pyramid  has  three  faces  above  lying  ex- 
actly over  the  three  faces  below.  It  is  symmetrically  placed 
in  reference  to  the  axes. 

The  trigonal  trapezohedron  has  three  faces  above  that 
do  not  lie  exactly  over  the  three  faces  below,  nor  do  they 
occur  in  alternating  position  as  is  the  case  with  the  rhom- 
bohedron. They  may  be  recognized  by  the  unsymmetrical 
position  of  each  face  with  reference  to  the  axes,  or  with 
reference  to  the  prism,  which  is  usually  present. 

The  ditrigonal prism  has  six  faces  that  make  alternately 
sharper  and  blunter  vertical  edges. 

In  case  the  trigonal  trapezohedron  is  present  a  crystal 
may  be  recognized  as  tetartohedral  by  the  fact  that  no 
symmetry  plane  whatever  is  present. 

II.    RHOMBOHEDRAL  TETARTOHEDRISM. 

The  forms  in  this  division  are  by  no  means  so  impor- 
tant as  are  those  in  the  foregoing  division,  and  are  repre- 
sented in  nature  by  only  a  few  not  very  common  minerals. 
A  very  brief  summary  of  the  possible  forms,  therefore,  will 
suffice. 

Basal  pinacoid  gives .Basal  pinacoid. 

Pyramid  of  first  order  gives. -Rhombohedron  of  first  order. 

Prism  of  first  order  gives Prism  of  first  order. 

Pyramid  of  second  order  gives 

Rhombohedron  of  second  order. 

Prism  of  second  order  gives Prism  of  second  order. 

Dihexagonal  pyramid  gives  Rhombohedron  of  third  order. 
Dihexagonal  prism  gives Prism  of  third  order. 


—  25  — 

The  prism  of  the  third  order  differs  from  the  prisms  of 
the  first  and  second  orders  only  in  its  position,  it  being  un- 
sym metrical  with  reference  to  the  crystal  axes.  Similarly, 
the  third  order  rhombohedron  has  an  unsymmetrical  posi- 
tion, but  otherwise  does  not  differ  from  the  other  rhom- 
bohedrons. 

The  symbols  of  tetartohedral  forms  may  be  expressed 
in  the  form  of  a  fraction  with  the  usual  holohedral  symbols 
for  numerators  and  4  for  the  denominator. 

For  a  further  description  of  the  symbols  as  well  as  for 
fuller  descriptions  of  these  tetartohedral  forms  the  reader  is 
referred  to  more  extended  treatises  on  crystallography.* 


HEMIMORPHIC  HEXAGONAL  FORMS. 

Hemimorphic  forms  are  half-forms  that  differ  from 
hemihedral  forms  in  a  very  important  respect,  namely,  the 
opposite  ends  of  some  (holohedral)  symmetry  axis  are  cut 
by  different  planes.  Usually  some  form  or  forms  that  are 
present  on  one  end  are  wanting  at  the  other  end  of  the 
axis. 

In  the  hexagonal  system  hemimorphic  forms  play  a 
very  important  and  interesting  role  in  the  case  of  a  very 
common  mineral,  tourmaline.  Here  we  have  the  hemi- 
morphism  superimposed  upon  a  rhombohedral  hemihe- 
drism.  This  gives  a  sort  of  quarter-form.  The  forms  that 
commonly  occur  hemimorphic  in  this  mineral,  i.  e.,  differ- 
ently developed  at  the  two  ends  of  the  vertical  axis,  are 
rhombohedrons,  scalenohedrons  and  the  basal  pinacoid.  In 
addition  to  these  there  are  usually  to  be  seen  a  trigonal 
prism  of  the  first  order,  a  ditrigonal  prism  and  the  prism  of 
the  second  order. 

These  forms  differ  from  tetartohedral  forms  in  that  we 
have  a  trigonal  prism  in  combination  with  a  scalenohedron, 
and  by  the  fact  that  the  trigonal  prism  is  of  the  first  order, 
instead  of  the  second  order.  The  trigonal  prism,  therefore, 


*  See  Elements  of   Crystallography,  by  George  H.  Williams,   New 
York,  Henry  Holt  &  Co.,  1890. 


—  26  — 


lies  under  the  rhombohedron,  i.  e.,  makes  a  horizontal  edge 
with  the  rhombohedron. 

The  explanation  of  the  trigonal  prism  is  as  follows  : 
The  hexagonal  prism  of  the  first  order  may  be  considered 
to  be  the  equivalent  of  a  rhombohedron  with  infinite  c  axis, 
three  faces  belonging  to  the  upper  and  three  to  the  lower 
half  of  the  crystal.  If,  now,  the  crystal  is  hemimorphic,  the 
three  prism  faces  that  belong  to  one  end  of  the  crystal  may 
be  developed  while  the  three  belonging  to  the  other  end 
may  be  suppressed.  In  the  same  way  the  ditrigonal  prism 
may  in  this  case  be  considered  to  be  the  hemimorphic  form 
of  a  scalenohedron  with  infinite  c  axis. 

TABLE    OF   SYMBOLS    IN     THE    HEXAGONAL   SYSTEM. 


WBIS9. 

NAD- 

DANA. 

MILLER- 

MANN. 

BRAVAIS. 

First  order  pyramid  . 

a 

a 

ooa 

:   me 

mP 

m 

(hohi) 

Second  order  pyramid 

2a 

a 

2a 

:  me 

mP2 

m-2 

(hh2h2i) 

Dihcxagonal  pyramid 

pa 

a 

na 

:   me 

mPn 

m-n 

(hkli) 

First  order  prism  

a 

a 

ooa 

:  ooc 

OOP 

I 

(10lO) 

Second  order  prism  .  .    2a 

a 

2a 

:  ooc 

coP2 

i-2 

(1020) 

Dihexagonal  prism.  . 

pa 

a 

na 

:  coc 

oo  Pn 

i-n 

(khlb) 

Basal   pinacoid. 

ooa 

ooa 

ooa 

:       c 

OP 

O 

(0001) 

ORTHORHOMBIC     SYSTEM. 

This  system  belongs  to  the  class  with  no  principal 
symmetry  plane.  It  has,  however,  three  common  symmetry 
planes  at  right  angles  to  each  other.  There  are,  there- 
fore, no  interchangeable  symmetry  axes. 

In  accordance  with  the  rule  for  the  selection  of  crystal 
axes  the  three  common  symmetry  axes  become  the  crystal 
axes.  None  of  these  axes  is  pre-eminent  above  the  others, 
as  there  is  no  principal  symmetry  axis.  We  may  select, 
then,  any  one  of  the  three  axes  for  the  vertical  axis,  the  two 
others  becoming  the  horizontal  axes.  The  shorter  of  the 
horizontal  axes  is  placed  from  front  to  back  and  is  called 


—  27  — 

the  brachy  (short)  axis.  It  is  designated  by  the  letter  a. 
The  longer  horizontal  axis  runs,  therefore,  from  right  to 
left,  and  is  called  the  macro  (long)  axis.  It  is  designated 
by  the  letter  b.  The  vertical  axis  is  designated  by  the  let- 
ter c. 

In  this  and  the  following  systems  the  unit  values  for 
the  three  axes  are  different,  and  the  ratio  between  them  is 
an  irrational  one.  On  the  other  hand  the  co-efficient  values 
are  always  simple  rational  quantities. 

CRYSTAL  FORMS  IN  THE  ORTHORHOMBIG  SYSTEM. 

There,  are  three  kinds  of  forms  in  this  system. 

First.    Forms  with  eight  faces Pyramids. 

Second.    Forms  with  four  faces Prism  and  Domes. 

Third.    Forms  with  two  faces..  __Pinacoids. 


Pyramid na 

Prism na 

Macro-dome a 

Brachy-dome GO  a 

Macro-pinacoid U 

Brachy-pinacoid oca 


b 

b 

cob 
b 

cob 
b 

GO  b 


me 


coc 
me 


me 


coc 
coc 


c 


Basal  pinacoid ooa 

Pyramids  cut  all  three  axes.  Prisms  and  domes  cut 
two  axes  and  are  parallel  to  one.  Pinacoids  cut  but  one 
and  are  parallel  to  two  axes. 

The  relative  lengths  of  any  two  axes  cannot  be  determined 
by  the  thickness  of  a  crystal,  but  only  by  means  of  some  face  which, 
if  extended,  would  cut  the  two  axes  in  question.  E.g.,  the  prism 
face  which  cuts  the  two  horizontal  axes  determines  which  is 
the  longer  and  which  is  the  shorter.  The  dome  determines 
the  relative  lengths  of  the  vertical  and  one  of  the  horizontal 
axes.  The  pyramid  face,  which  cuts  all  three  axes  determ- 
ines the  relative  lengths  of  the  three  axes. 

It  is  evident  that  the  prism  and  the  domes  are  virtually 
the  same  thing  with  different  names,  as  we  can  change 
either  dome  into  a  prism  by  a  different  selection  of  the  ver- 
tical axis.  In  a  certain  sense,  therefore,  the  domes  and  the 
prism  are  interchangeable.  In  the  same  sense  the  three 
pinacoids  are  also  interchangeable. 


—  28  — 

RULES    FOR      DETERMINING    FORMS    IN    THE    ORTHORHOMBIC 
SYSTEM. 

First.  A  face  whose  plane  cuts  all  three  axes  belongs 
to  a  pyramid. 

Second.  A  face  parallel  to  one  axis  belongs  either  to 
a  prism  or  to  a  dome ;  if  parallel  to  the  vertical  axis  it 
belongs  to  a  prism ;  if  parallel  to  the  brachy  (short)  axis,  to 
a  brachy-dome;  if  parallel  to  the  macro  (long)  axis,  to  a 
macro-dome. 

Third.  A  face  parallel  to  two  axes  belongs  to  a  pina- 
coid ;  if  parallel  to  the  vertical  and  brachy  axes  it  belongs 
to  the  brachy-pinacoid ;  if  parallel  to  the  vertical  and  macro 
axes,  to  the  macro-pinacoid  ;  if  parallel  to  the  two  horizontal 
axes,  to  the  basal  pinacoid. 

Fourth.  There  may  occur  on  the  same  crystal  an  in- 
definite number  of  pyramids  or  of  prisms  or  domes,  because 
these  forms  have  variable  symbols ;  but  the  three  pinacoids 
can  occur  but  once,  as  their  symbols  are  invariable. 


HEMIHEDRAL  AND  HEMIMORPHIG  FORMS  IN  THE  ORTHORHOMBIC 
SYSTEM. 

There  is  but  one  kind  of  hemihedrism  possible  in  this 
system,  namely,  the  one  corresponding  to  the  inclined  in 
the  isometric  system,  and  to  the  sphenoidal  in  the  tetra- 
gonal system.  This  is  produced  by  cutting  the  holohedral 
forms  by  means  of  the  common  symmetry  planes  into 
octants,  and  by  suppressing  the  faces  that  lie  wholly  within 
alternating  octants  while  the  remaining  faces  are  developed. 

There  is  only  one  form  that  can  produce  a  new  form, 
and  this  is  the  pyramid.  This  produces  the  orthorhombic 
sphenoid,  which  is  similar  to  the  tetragonal  sphenoid,  but 
without  interchangeable  axes. 

Hemimorphic  forms  are  fairly  common  in  this  system. 
They  are  developed  by  suppressing  certain  forms  at  one 
end  of  a  symmetry  axis  that  are  developed  at  the  other  end. 


—  29  — 
TABLE   OF   SYMBOLS    IN    THE   ORTHORHOMBIC   SYSTEM. 


WEISS. 

NAUMANN. 

DANA. 

MILLER. 

Pyramid  

na 

b 

me 

mPn 

m-n 

(hkl) 

Prism 

na 

b 

ooc 

mP 

I 

(110) 

Brachy-dome  
Macro-dome  

ooa 
a 

b 

QO  b 

me 
me 

oo  Pn 
oo  Pn 

i-« 
i-n 

(khO)  h  >  k 
(hkO)  h  >  k 

Brachy-pinacoid  . 
Macro-pinacoid  .  . 
Basal  pinacoid  

ooa 
a 
ooa 

b 
cob 

QO  b 

ooc 

OOC 

c 

ooP& 

QOPOO 

OP 

i.r 

i-T 
O 

(010) 
(100) 
(001) 

MONOCLINIC    SYSTEM. 

In  this  system  there  is  but  one  symmetry  plane  and, 
therefore,  but  one  symmetry  axis. 

In  accordance  with  the  rule  we  select  this  symmetry 
axis  for  one  of  the  crystal  axes.  It  is  the  only  natural 
crystal  axis. 

The  two  remaining  axes  are  selected  to  lie  in  the 
symmetry  plane  and  parallel  to  prominent  edges,  or,  if  no 
edge  is  available,  parallel  to  a  prominent  face.  (Only  in 
rare  exceptions  must  a  line  connecting  two  corners  be 
chosen).  If  an  axis  be  parallel  to  an  edge  it  must  neces- 
sarily be  parallel  to  the  faces  forming  the  edge. 

Now,  as  edges  parallel  to  the  symmetry  plane  in  the 
monoclinic  system  are  not  found  to  occur  exactly  at  right 
angles  to  each  other,  the  two  axes  in  the  symmetry  plane 
are  also  never  at  right  angles. 

We  have,  then,  two  axes  lying  in  the  symmetry  plane 
oblique  to  each  other,  and  one  axis  coinciding  with  the 
symmetry  axis,  and,  therefore,  at  right  angles  to  the  other  two 
axes. 

In  orienting  the  crystal  the  symmetry  plane  is  placed 
vertical  and  from  front  to  back,  so  that  the  symmetry  axis 
becomes  the  b  axis,  and  is  called  the  ortho-axis.  The 
crystal  is  then  turned  until  one  of  the  oblique  axes  becomes 
vertical  and  the  other  slopes  downward  from  the  center  of 


Of) 
Ov 

the  crystal  towards  the  observer,  i,  e.,  towards  the  front. 
The  vertical  axis  then  becomes  the  c  axis ;  and  the  other,  or 
clino-axis  the  a  axis. 

CRYSTAL  FORMS  IN  THE  MONOCLINIC  SYSTEM. 

First.     Forms    with    four  faces — prisms,   clino-domes 
and  pyramids. 

Second.     Forms   with    two    faces — ortho-domes    and 
pinacoids. 

f_  (Positive..        _  -na 


First 


I  Pyramids   J  Negative '_"_""  +na 

i  Prism na 

^Clino-dome ooa 

Positive -a 

Negative +a 

Second.  J  Ortho-pinacoid a 

I  Basal  pinacoid ooa 

^Clino-pinacoid oca 


Ortho-domes 


b 
b 
b 
b 

oob 
oob 
oob 

QO  b 
b 


me 
me 


ooc 
me 


me 
me 


ooc 
c 


QOC 


The  plane  in  which  lie  the  vertical  and  the  ortho-axes 
divides  the  crystal  into  two  unsymmetrical  parts.  There- 
fore, the  faces  that  may  occur  on  one  side  of  this  plane  are 
not  repeated  on  the  other  side,  except  that  every  face  must 
have  an  opposite  parallel  face.  For  instance,  a  face  in  the 
pyramid  position  cutting  the  vertical  and  ortho-axes  and 
the  front  end  of  the  clino  axis  need  not  occur  at  the  rear  of 
the  crystal  in  a  symmetrical  position.  The  two  pyramid 
faces  in  front,  above  the  basal  pinacoid,  together  with  the 
two  opposite  and  parallel  faces  at  the  back,  below  the  basal 
pinacoid,  make  up  a  monoclinic  pyramid  which  is  called  a 
partial  pyramid.  It  takes  two  of  these  partial  pyramids  to 
correspond  with  the  orthorhombic  pyramid. 

For  a  similar  reason  there  are  two  partial  ortho- domes. 
In  the  case  of  these  partial  forms  if  a  face  cuts  the  ends  of 
the  vertical  and  clino  axes  that  form  an  obtuse  angle  it 
belongs  to  a  negative  partial  form.  If  it  cuts  the  ends  of 
these  axes  forming  the  acute  angle  the  form  is  positive. 

The  pyramid,  prism  and  clino-dome  are  virtually  the 
same  thing,  inasmuch  as  they  can  be  changed  the  one  into 
the  other  by  changing  the  locations  of  the  two  oblique  axes. 


—  31  — 

For  the  same  reason  the  basal  pinacoid,  ortho-pinacoid 
and  ortho-dome  may  be  changed  the  one  into  the  other. 

The  clino-pinacoid  is  the  only  form  that  cannot  be 
changed  into  an  other  by  changing  the  position  of  the 
oblique  axes,  for  the  reason  that  it  is  the  only  form  parallel 
to  a  symmetry  plane. 


RULES  FOR  DETERMINING  FORMS  IN  THE  MONOCLINIG  SYSTEM. 

First.  A  face  whose  plane  cuts  all  three  axes  belongs 
to  a  pyramid.  If  it  lies  over  the  acute  angle  of  the  oblique 
axes  it  belongs  to  a  positive  pyramid ;  if  over  the  abtuse 
angle,  to  a  negative  pyramid. 

Second.  A  face  parallel  only  the  vertical  axis  belongs 
to  a  prism. 

Third.  A  face  parallel  only  the  clino-axis  belongs  to 
a  clino-dome. 

Fourth.  A  face  parallel  only  to  the  ortho-axis  belongs 
to  an  ortho-dome. 

Fifth.  A  face  parallel  to  two  axes  is  a  pinacoid ;  if  it 
is  parallel  to  the  clino  and  the  ortho-axes  it  belongs  to  the 
basal  pinacoid ;  if  to  the  vertical  and  the  clino-axes,  to  the 
clino-pinacoid  ;  if  parallel  to  the  vertical  and  the  ortho- 
axes,  to  the  ortho-pinacoid. 

Sixth.  All  the  forms  except  the  three  pinacoids  may 
occur  an  indefinite  number  of  times  on  the  same  crystal 
because  they  have  variable  symbols. 


HEMIHEDRAL  AND  TETARTOHEDRAL  FORMS    IN  THE  MONOCLINIG 
SYSTEM. 

Tetartohedral  forms  are  not  certainly  known  to  occur. 
Hemihedral  forms  are  known  to  exist,  but  their  occurrence 
is  so  rare  that  a  discussion  of  these  forms  is  considered 
beyond  the  scope  of  these  lecture  notes. 


—  32  — 
TABLE   OF   SYMBOLS    IN    THE    MONOCLINIC   SYSTEM. 


WKJSS. 

NAUMANN. 

DANA. 

MILLER 

Pyramid  j  Positive.. 
(  Negative... 
Prism         ._ 

-na 
+na 
,     na 
oca 

b 
b 
b 
b 

me 
me 

DOC 

me 

+  mPn 
-mPn 
ocP 
mP^o 

+  m-n 
-m-n 
I 
m-> 

(hkl) 
(hkl) 
(110) 
(Okl) 

Clino-dome 

Ortho-dome  j  P°sitive 
(  Negative 

Orthopioacoid    ..   . 

-a 
+a 
a 

ooa 

oob 

GO  b 
oob 

oo  b 

me 
me 
ooc 
c 

+  mPoB 
-mPo> 
ooPoo 

OP 

+  m-7 
-m-T 

i-i 

O 

(hOl) 
(hOl) 
(100) 
(001) 

Basal  pinacoid 

Clino  pinacoid..  

QO  a 

b 

COC 

GO  POO 

ixi 

(010) 

TRICLINIC    SYSTEM. 

In  this  system,  as  there  is  no  symmetry  plane,  and, 
therefore,  no  symmetry  axis,  there  can  also  be  no  natural 
crystal  axes.  In  selecting  the  crystal  axes,  it  is  customary 
to  chose  directions  parallel  to  prominent  edges  on  the 
crystal,  seeking  at  the  same  time  to  secure  axes  as  near 
right  angles  as  the  crystal  will  allow.  (Sometimes  it  will 
be  necessary  to  chose  a  direction  parallel  to  two  faces  that 
do  not  meet  in  an  edge,  or  a  diagonal  through  the  crystal 
connecting  two  corners.) 

As  edges  and  faces  in  the  triclinic  system  are  never 
exactly  at  right  angles,  although  approximately  so,  we  can- 
not have  crystal  axes  at  right  angles. 

As  is  the  case  in  the  orthorhombic  system,  we  choose 
any  one  of  the  three  axes  for  the  vertical  axis,  place  the 
longer  of  the  other  two  from  right  to  left  and  the  third, 
therefore,  as  nearly  from  front  to  back  as  the  obliquity  of 
the  axes  will  allow.  The  shorter  axis,  a,  is  called  the 
brachy-axis ;  the  longer,  b,  the  macro-axis,  exactly  as  in 
the  orthorhombic  system. 

Each  form  in  this  system  consists  of  but  pairs  of 
parallel  planes.  There  is  no  essential  difference  between 


—  33  — 

them,  as  any  form   may  be   changed   into  any  other   form 
merely  by  making  a  different  selection  of  the    crystal  axes. 
Forms    are  named  as  they    are  in  the    orthorhombic 
system. 

RULES  FOR  DETERMINING    CRYSTAL    FORMS    IN    THE  TRIGLINIO 
SYSTEM. 

First.  All  faces  whose  planes  cut  all  three  axes  belong 
to  pyramids.  There  are  four  cf  these  partial  pyramids  to 
two  in  the  monoclinic  and  to  one  in  the  orthorhombic 
systems. 

Second.  Faces  parallel  only  to  the  vertical  axis  belong 
to  prisms.  There  are  two  of  these  partial  prisms. 

Third.  Faces  parallel  only  to  the  brachy-axis  belong 
to  brachy-domes.  There  are  two  of  these  partial  brachy  - 
domes. 

Fourth.  Faces  parallel  only  to  the  macro-axis  belong 
to  macro- domes.  There  are  two  of  these  partial  macro- 
domes. 

Fifth.  Faces  parallel  to  the  brachy  and  macro-axes 
belong  to  the  basal  pinacoid. 

Sixth.  Faces  parallel  to  the  vertical  and  brachy-axes 
belong  to  the  brachy-pinacoid, 

Seventh.  Faces  parallel  to  the  vertical  and  macro- 
axes  belong  to  the  macro-pinacoid. 

Eighth.  The  pyramids,  prisms,  brachy  and  macro- 
domes,  having  variable  symbols,  may  occur  an  indefinite 
number  of  times  on  a  crystal ;  all  the  other  forms  but  once. 

SYMBOLS. 

The  symbols  used  are  exactly  the  same  as  those  given 
for  the  orthorhombic  system.  To  distinguish  between  the 
different  partial  forms  special  accents  and  positive  and 
negative  signs  are  used. 


—  34  — 

DISTORTION  OF  CRYSTALS. 

In  all  the  foregoing  cases  we  have  had  under  considera- 
tion ideally  developed  crystals  or  crystal  models.  In 
nature,  however,  perfect  crystals  are  very  rare.  Almost  in- 
variably they  are  to  some  extent  distorted.  Except  in  the 
case  of  mechanical  distortion  this  crystal  distortion  is 
usually  of  such  a  nature  that  the  inclinations  of  the  faces  to 
each  other  and  to  the  axes  remain  unaltered.  This  may  be 
conceived  to  be  accomplished  by  the  shoving  of  one  or 
more  faces  parallel  to  themselves  so  that  some  faces  are  in- 
ordinately developed  at  the  expense  of  other  faces.  In  this 
way  some  faces  may  be  completely  crowded  off  the  crystal 
with  an  apparent  loss  of  symmetry. 

In  natural  crystals,  therefore,  we  cannot  recognize 
symmetry  planes  by  the  symmetrical  development  of  faces 
on  both  sides  of  a  plane,  but  merely  by  the  equal  inclina- 
tions of  corresponding  faces  to  the  symmetry  plane. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
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OVERDUE. 


1940 


APR    B 


I940 


MAY 


'940 


MAY  i9 194 


M^l/  1  *-t  HQAH  M 

i 

LD  21-100m-7,  '39(402    / 

crystallc 


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